Optimal quantum state determination by constrained elementary measurements

The purpose of this short note is to utilize the work on isotropic lines, described by Albouy [J. Phys. A. Math. Theor., vol. 42 (2009), 072001], on Wigner distributions for finite-state systems as described by Chaturvedi et al. [J. Phys. A. Math. Theor., vol. 43 (2010), 0753075302], estimation of the state of a finite level quantum system based on Weyl operators in the $L^2$-space over a finite field as described by Parthasarathy in [Inf. Dimens. Anal. Quantum Prob. Relat. Top., Vol. 07, Issue 4, Dec. 2004. 607-617] to display maximal abelian subsets of certain unitary bases for the matrix algebra $M_d$ of complex square matrices of order $d>3$; and then, combine these special forms with constrained elementary measurements to obtain optimal ways to determine a $d$-level quantum state. This enables us to generalise illustrations and strengthen results related to quantum tomography by Ghosh and Singh in [arXiv:1401.0099v1 [quant-ph] 31 Dec 2013].